A moving coil galvanometer coil has a coil of area A, number of turns N. The radial magnetic field present is B. the moment of inertia of the coil is I about its rotation axis. The torque is applied by the magnetic field on the coil of the galvanometer when current `I_(0)` passes through it and produces a deflection of `pi//2` of the pointer. Then answer the following questions based on the paragraph.
If the charge Q is passed almost instantaneosly through the coil, the angular velocity of the coil, immediately after this is

Options

A. NIBAQ
B. `(NBAQ)/(I)`
C. `(2NBAQ)/(I)`
D. `(NBAQ^(2))/(I)`

Answer

To find the angular velocity of the coil immediately after a charge \( Q \) is passed through it, we can follow these steps:

### Step 1: Understand the Torque on the Coil
The torque \( \tau \) acting on the coil due to the magnetic field when a current \( I_0 \) flows through it is given by the formula:
\[ \tau = N I_0 A B \]
where:
- \( N \) = number of turns in the coil
- \( A \) = area of the coil
- \( B \) = magnetic field strength

### Step 2: Relate Torque to Angular Acceleration
The torque is also related to the moment of inertia \( I \) and angular acceleration \( \alpha \) by the equation:
\[ \tau = I \alpha \]
From this, we can express angular acceleration as:
\[ \alpha = \frac{\tau}{I} \]

### Step 3: Substitute Torque into the Angular Acceleration Equation
Substituting the expression for torque from Step 1 into the equation for angular acceleration gives:
\[ \alpha = \frac{N I_0 A B}{I} \]

### Step 4: Relate Charge to Current
The current \( I_0 \) can be expressed in terms of charge \( Q \) and time \( t \) as:
\[ I_0 = \frac{Q}{t} \]
Substituting this into the angular acceleration equation yields:
\[ \alpha = \frac{N \left(\frac{Q}{t}\right) A B}{I} = \frac{N Q A B}{I t} \]

### Step 5: Find Angular Velocity Immediately After Charge is Passed
The angular velocity \( \omega \) can be found using the relationship between angular acceleration and time. Since the charge is passed almost instantaneously, we can consider the time \( t \) to be very small, leading to:
\[ \omega = \alpha \cdot t \]
Substituting the expression for \( \alpha \):
\[ \omega = \left(\frac{N Q A B}{I t}\right) \cdot t = \frac{N Q A B}{I} \]

### Final Result
Thus, the angular velocity \( \omega \) of the coil immediately after the charge \( Q \) is passed through it is:
\[ \omega = \frac{N Q A B}{I} \]

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